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Revisiting extensions of regularly varying functions
Meitner Cadena
To cite this version:
Meitner Cadena. Revisiting extensions of regularly varying functions. 2015. �hal-01105176v2�
Revisiting extensions of regularly varying functions
Meitner Cadena
∗Abstract
Relationships among the classesM,M∞, andM−∞and the class ofO-regularly varying functions are shown. These results are based on two characterizations ofM,M∞, and M−∞provided by Cadena and Kratz in [7] and a new one given in this note.
Keywords: regularly varying function; slowly varying function; O-RV; large deviations;
extreme value theory
AMS classification: 26A12; 60F10
1 Introduction
A positive and measurable functionUdefined onR+is aregularly varying(RV) function if
x→∞lim U(t x)
U(x) < ∞ (t>0). (1)
If this limit equals 1,Uis aslowly varying(SW) function. ClassesRV andSV of regularly and slowly varying functions were introduced by Karamata [10] in 1930. Since then theory of these functions has been developed in many directions. Systematic treatment of this theory can be found in e.g. [6] and [14].
Extensions of RV functions have been obtained by letting (1) to vary. An early extension of this type was given by Avakumovi´c in 1936 [4]. He introduced the classO-RVofO-regularly varying (O-RV) functionsUwhich satisfy the following condition instead of (1):
0<U∗(t) := lim
x→∞
U(t x) U(x) ≤ lim
x→∞
U(t x)
U(x) =:U∗(t)< ∞ (t≥1). (2) Recently Cadena and Kratz [7] gave an extension of RV functions by also letting (1) to vary, but they designed it in a different way to the previous one. They introduced the classM which consists in functionsUsatisfying the following condition instead of (1):
∃ρ∈R,∀ǫ>0, lim
x→∞
U(x)
xρ+ǫ=0 and lim
x→∞
U(x)
xρ−ǫ = ∞. (3)
We have clearlyRV (O-RV and, for instance using Theorem 1 (see Corollary 1), RV(M. There arises the natural question of howO-RVandM are related between them. We under- take this study helping us of characterizations of these classes: recalling well-known charac- terizations ofO-RVand giving proofs of three characterizations ofM, two of them provided in [7] and a new one given in this note.
∗UPMC Paris 6 & CREAR, ESSEC Business School; E-mail: meitner.cadena@etu.upmc.fr, b00454799@essec.edu, or meitner.cadena@gmail.com
Cadena and Kratz also introduced the following natural extensions ofM. M∞ :=
½
U:R+→R+:Uis measurable and satisfies∀ρ∈R, lim
x→∞
U(x) xρ =0
¾ (4) M−∞ :=
½
U:R+→R+:Uis measurable and satisfies∀ρ∈R, lim
x→∞
U(x) xρ = ∞
¾ . (5) The new characterization given forM is extended toM∞andM−∞. Relationships among M∞andM−∞andO-RVare also investigated in this note.
This note is organized as follows. The main results are presented in the next section, intro- ducing previously notations and definitions. First, the new characterizations ofM,M∞, and M−∞based on limits are given. Next, analyses of uniform convergence in these characteriza- tions are presented and, finally, relationships amongO-RVandM,M∞andM−∞are shown.
All proofs are collected in Section 3. Conclusion is presented in the last section.
2 Main Results
For a positive functionUwith supportR+itslowerandupper ordersare defined by (see e.g.
[6])
µ(U) := lim
x→∞
log (U(x))
log(x) , ν(U) := lim
x→∞
log (U(x)) log(x) . Throughout this note log(x) represents the natural logarithm ofx.
We notice that the classesM,M∞, andM−∞defined in (3), (4), and (5) are a bit weaker than the corresponding classes given in [7], and that each of them is disjoint from each other. More- over, using straightforward computations, one can prove thatρdefined in (3) is unique, hence it will be denoted byρU, and one can show thatǫ>0 in (3) can be taken sufficiently small.
Additionally, one can prove thatM is strictly larger than RV, for instance using Theorem 1 (see Corollary 1), and thatM∞is related to the domain of attraction of Gumbel (see [7]).
The new characterizations ofM,M∞, andM−∞follow.
Theorem 1. Let U:R+→R+be a measurable function. Then (i) U∈MwithρU= −τiff
∀r<τ,∃xa>1,∀x≥xa, lim
t→∞trU(t x) U(x) =0
∀r>τ,∃xb>1,∀x≥xb, lim
t→∞trU(t x) U(x) = ∞.
(6)
(ii) U∈M∞iff
∀r∈R,∃x0>1,∀x≥x0, lim
t→∞trU(t x)
U(x) =0. (7)
(iii) U∈M−∞iff
∀r∈R,∃x0>1,∀x≥x0, lim
t→∞trU(t x)
U(x) = ∞. (8)
Example1.
1. Consider a measurable and positive function U with supportR+such that, for x≥x0with some x0>1, U(x)=x±
log(x).
Noting that, for t,x>1,
t→∞limtrU(t x) U(x) =lim
t→∞tr−1 log(x) log(t x)=
½ 0 if r>1
∞ if r<1, provides, takingτ= −1and applying Theorem 1, (i), U∈MwithρU=1.
2. Let U be a function defined by U(x) :=xsin(x), x>0.
Writing
trU(t x)
U(x) =tr+sin(t x)xsin(t x)−sin(x)
gives, for r∈R,
t→∞limtrU(t x)
U(x) = ∞ and lim
t→∞trU(t x) U(x) =0.
Hence the necessary condition of Theorem 1, (i), is not satisfied and consequently gives U6∈M.
It follows a consequence of Theorem 1. This result was proved by Cadena and Kratz in [7]
combining a result provided in [8] and another characterization ofM (see Theorem CK later).
Corollary 1. RV (M.
Note that, from Corollary 1,RV⊆MTO-RV.
There are not common elements betweenO-RVandMunder their definitions given in (2) and (3) respectively, but observing the characterization ofMgiven in Theorem 1 one identifies the quotientU(t x)±
U(x), which appears in (2). The next example exploits this link to show a first relationship betweenO-RVandM.
Example2. M 6⊆ O-RV.
Let U be a function defined by U(x) :=exp©
(logx)αcos¡
(logx)β¢ª
, x>0, where0<α,β<1such thatα+β>1.
Prof. Philippe Soulier gave recommendations to correct an error in an early version of this ex- ample.
On the one hand, noting that, for x,t>e, using the changes of variable y=log(x)and s=log(t) and observing that s→ ∞as t→ ∞,
t→∞limtrU(t x) U(x) =lim
s→∞expn
r s+(s+y)αcos¡
(s+y)β¢
−yαcos¡ yβ¢o
=lim
s→∞exp
½ s
µ r+ 1
s1−α
³ 1+y
s
´α
cos¡
(s+y)1/3¢
−yα s cos¡
yβ¢
¶¾
=
½ 0 if r>0
∞ if r<0, provides, takingτ=0and applying Theorem 1, U∈MwithρU=0.
On the other hand, writing, for x>e and t>0, using the previous changes of variables, with x such that¡
logt x¢β
=π±
2+2kπ, for a given t, U(t x)
U(x) =expn
−¡ logx¢α
cos¡
(logx)β¢o
=expn
−yαcos¡ ((π±
2+2kπ)1/β−s)β¢o
=expn
¡(π±
2+2kπ)1/β−s¢α
sin¡ ((π±
2+2kπ)1/β−s)β−(π±
2+2kπ)¢o .
Since¡ (π±
2+2kπ)1/β−s¢β
−(π±
2+2kπ)→0as k→ ∞, we have
k→∞lim h¡
(π±
2+2kπ)1/β−s¢α
sin¡ ((π±
2+2kπ)1/β−s)β−(π±
2+2kπ)¢i
= lim
k→∞
((π±
2+2kπ)1/β−s)β−(π± 2+2kπ)
¡(π±
2+2kπ)1/β−s¢−α , which is an indetermination of type0±
0. Then, applying L’Hopital’s rule we have
klim→∞
((π±
2+2kπ)1/β−s)β−(π± 2+2kπ)
¡(π±
2+2kπ)1/β−s¢−α
= lim
k→∞(2π)α/β((π±
2+2kπ)1/β−s)β−(π± 2+2kπ) k−α/β
= lim
k→∞−β
α(2π)α/β+1((π±
2+2kπ)1/β−s)β−1(π±
2+2kπ)1/β−1−1
k−α/β−1 ,
which is an indetermination of type0±
0. Then, applying again L’Hopital’s rule we have
k→∞lim ((π±
2+2kπ)1/β−s)β−(π± 2+2kπ)
¡(π±
2+2kπ)1/β−s¢−α
= lim
k→∞sβ(1−β)
α(α+β)(2π)α/β+2((π±
2+2kπ)1/β−s)β−2(π±
2+2kπ)1/β−2 k−α/β−2
= lim
k→∞sβ(1−β)
α(α+β)(2π)(α+β−1)/βk(α+β−1)/β
=
½ ∞ if s>0
−∞ if s<0.
Then, we get, for t>1,
U∗(t)= lim
x→∞
U(t x) U(x) = ∞, and, for t<1,
U∗(t)= lim
x→∞
U(t x) U(x) =0, which contradict(2), so U6∈O-RV. In particular, U6∈SV.
Next, the uniform convergences inxof limits given in (6), (7), and (8) are analyzed. To this aim, we will use the next two results.
Proposition 1. Let U:R+→R+be a measurable function. Then
(i) If U∈M withρU= −τ, then there exists x0>1such that, for x0≤c<d< ∞, there exist 0<Mc<Mdsatisfying, for x∈[c;d], Mc≤U(x)≤Md.
(ii) If U∈M∞, then there exists x0>1such that, for c≥x0, there exist Mc>0satisfying, for x∈[c;∞), U(x)≤Mc.
(iii) If U∈M−∞, then there exists x0>1such that, for d≥x0, there exist Md>0satisfying, for x∈[d;∞), U(x)≥Md.
Proposition 2(Given in [2]). Letµbe the Lebesgue measure onR, A a measurable set of positive measure, and©
xnª
n∈Na bounded sequence of real numbers. Then,µ(A)≤µ¡
limn→∞(xn+A)¢ . Now the results on uniform convergences are presented. Their proofs are inspired by [3].
Theorem 2(Uniform Convergence Theorem (UCT)). Let U:R+→R+be a measurable func- tion. Then
(i) If U∈M withρU= −τand r<τ, then, for any xa≤c<d< ∞for some xa>1,
t→∞limtr sup
x∈[c;d]
U(t x) U(x) =0.
(ii) If U∈M withρU= −τand r>τ, then, for any xb≤c<d< ∞for some xb>1,
t→∞lim tr inf
x∈[c;d]
U(t x) U(x) = ∞.
(iii) If U∈M∞satisfying, for s>1, U(x)≥Msfor x∈[1;s]and some Ms>0, then, for r∈R and any constants x0≤c<d< ∞for some x0>1,
t→∞limtr sup
x∈[c;d]
U(t x) U(x) =0.
(iv) If U∈M−∞satisfying, for s>1, U(x)≤Msfor x∈[1;s]and some Ms>0, then, for r∈R and any constants x0≤c<d< ∞for some x0,
t→∞lim tr inf
x∈[c;d]
U(t x) U(x) = ∞.
Note that UCT cannot be extended to infinite intervals. For instance, from the functionU given in Example 2 we have that computing the supremum of the quotientU(t x)±
U(x) inxon [x0;∞), for anyx0>1, gives always∞, and hence one cannot deduce thatρU=0.
The next results onO-RV,M,M∞, andM−∞will be used to give more relationships between these classes. OnO-RVwe need:
Proposition 3(see e.g. [11], [14], [1], [9], and [6]). Let U:R+→R+be a measurable function.
Then the following statements are equivalent:
(i) U∈O-RV.
(ii) There existα,β∈Rand x0>1,c>0such that, for all t≥1and x≥x0, c−1tβ≤U(t x)
U(x) ≤ctα.
(iii) There exist functionsη(x)andφ(x)bounded on[x0;∞), for some x0≥1, such that U(x)=exp
½ η(x)+
Zx 1
φ(y)d y y
¾
, x≥1.
OnMwe need the next two characterizations ofMgiven by Cadena and Kratz in [7]. For the sake of completeness of this note, we give them as Theorem CK and indicate their proofs. Part of these proofs are copied from [7].
Theorem CK. Let U:R+→R+be a measurable function. Then the following statements are equivalent:
(i) U∈MwithρU=τ.
(ii) lim
x→∞
log (U(x)) log(x) =τ.
(iii) There exist b>1and measurable functionsα,β, andδsatisfying, as x→ ∞, α(x)±
log(x)→0, β(x)→τ, δ(x)→1, such that
U(x)=exp
½
α(x)+δ(x) Zx
b β(s)d s s
¾
, x≥x1for some x1≥b.
Remark 1. If F is the tail of a distribution F associated to a random variable (rv) X , some au- thors (see e.g. [12] and [13]) say that X is heavy-tailed if the limit
η:= lim
x→∞
log³ F(x)´ log(x) exists and takes a negative value.
We notice that this characterization does not cover rvs with heavy tails satisfyingη=0or with heavy tails for which such limit does not exist. Indeed, on the one side, from Theorem CK one has thatη=0implies that F∈M withρF=0, being a particular case of these functions the SV functions, which are considered heavy-tailed. On the other side, Cadena and Kratz presented in [7] families of tails F for which the limit lim
x→∞
log³ F(x)´
log(x) does not exist, for instance the next tail defined by (see [7])
Letα>0,β< −1, xa>1, and define the series xn=xa(1+α)n, n≥1, which satisfies xn→ ∞as n→ ∞. It is not hard to prove that the tail F associated to a rv X and defined by
F(x) :=
( 1 x∈[0;x1)
xα(1+β)n x∈[xn;xn+1),∀n≥1 satisfies
x→∞lim log³
F(x)´
log(x) = −α(1+β)
1+α < −α(1+β)= lim
x→∞
log³ F(x)´ log(x) . Note that if−α(1+β)±
(1+α)<1, then the expected value of X is∞, which means that X can be considered as a heavy-tailed rv.
We notice from the representations ofUviaO-RVandMgiven in Proposition 3, (iii), and The- orem CK, (iii), respectively, that a key difference between those representations is the presence of a bounded function under the integral symbol. Motivated by this observation, we built the next function belonging toO-RV but not toM. This aim is reached by building a bounded functionφsuch that the limit lim
x→∞
Rx 1φ(s)d ss
log(x) does not exist. Note that if this limit exists, then, applying Theorem CK, (iii),U∈M.
Example3. O-RV 6⊆ M.
Let U:R+→R+be a measurable function satisfying, for x≥1, U(x)=exp
½Zx
1
φ(s)d s s
¾ , where the functionφhas support[1;∞)and is defined by
φ(x)=
½ 0 if x∈[1;e)or x∈Inwith n odd 1 if x∈Inwith n even,
where In=[een;een+1), n∈N.
On the one hand, applying Proposition 3, one has U∈O-RV.
On the other hand, writing, for x>1, using the change of variable y=log(s)± log(x), log(U(x))
log(x) = Rx
1φ(s)d ss log(x) =
Z1
0 φ³ eylog(x)´
d y gives, taking xn=een, n=2, 3, . . .,
log(U(xn)) log(xn) =
n−1
X
k=1
Zek+1/en
ek/en φ³ ey en´
d y=
n−1
X
k=0
(−1)ke−k if n is odd
n−1X
k=1
(−1)k+1e−k if n is even, and one then gets
n→∞lim
log(U(xn)) log(xn) =
1
1+e−1 if n is odd e−1
1+e−1 if n is even, which impliesν(U)−µ(U)≥(1−e−1)±
(1+e−1)>0, hence the limit lim
x→∞
log(U(x))
log(x) does not exist and thus, applying Theorem CK, U6∈M.
Now we give another relationship betweenO-RVandM.
Proposition 4. Let U:R+→R+be a measurable function. If U∈O-RV and the limit
x→∞lim
log(U(x)) log(x) exists, then U∈M.
The relationships ofM∞andM−∞withO-RVare simpler.
Proposition 5. Forλ∈©
∞,−∞ª ,MλT
O-RV= ;.
3 Proofs
Proof of Theorem 1.
• Proof of the necessary condition of (i)
AssumeU∈MwithρU= −τ. Letr∈Rsuch thatr6=0.
– If r<τ
Let 0<ǫ<τ−randδ>0. By hypothesis, there exists a constantx0>1 such that, for x≥x0,U(x)≤δx−τ+ǫ, and there existsx1>1 such that, forx≥x1,U(x)≥x−τ−ǫ±
δ.
Hence, settingxa:=max(x0,x1), forx≥xaandt>1, trU(t x)
U(x) ≤δ2tr(t x)−τ+ǫxτ+ǫ=δ2t−τ+r+ǫx2ǫ, and the assertion then follows ast→ ∞since−τ+r+ǫ<0.
– If r>τ
Let 0<ǫ<r−τandδ>0. By hypothesis, there exists a constantx0>1 such that, for x≥x0,U(x)≤δx−τ+ǫ, and there existsx1>1 such that, forx≥x1,U(x)≥x−τ−ǫ±
δ.
Hence, settingxa:=max(x0,x1), forx≥xaandt>1, trU(t x)
U(x) ≥ 1
δ2tr(t x)−τ−ǫxτ−ǫ= 1
δ2tr−τ−ǫx−2ǫ, and the assertion then follows ast→ ∞sincer−τ−ǫ>0.
• Proof of the sufficient condition of (i) Letδ>0 andη>0.
One the one hand, sinceτ−δ±
2<τ, by hypothesis, there exists a constantxa>1 such that, forx≥xa, lim
t→∞tτ−δ/2U(xt)
U(x) =0. Hence, givenx≥xa, there existsta=ta(x)>1 such that, fort≥ta,tτ−δ/2U(t x)≤ηU(x), or
U(t x)
(t x)−τ+δ≤ηxτ−δU(x)
tδ/2 . (9)
One the other hand, sinceτ+δ±
2>τ, by hypothesis, there exists a constantxb>1 such that, forx≥xb, lim
t→∞tτ+δ/2U(xt)
U(x) = ∞. Hence, givenx≥max(xa,xb), there existstb= tb(x)>1 such that, fort≥tb,tτ+δ/2U(t x)≥ηU(x), or
U(t x)
(t x)−τ−δ≥ηxτ+δU(x)tδ/2. (10)
Combining (9) and (10), givenx ≥max(xa,xb) and fort ≥max(ta,tb), and using the change of variabley=t xwithy→ ∞ast→ ∞, provide, forδ>0,
y→∞lim U(y)
y−τ+δ=0 and lim
y→∞
U(y) y−τ−δ= ∞, which implies thatU∈M withρU= −τ.
• Proof of the necessary condition of (ii)
Letr∈Randη>0. Setr′< −r. SinceU∈M∞there exists a constantx0>1 such that, forx≥x0,U(x)≤ηxr′. Hence, fort>1,
trU(t x)
U(x) ≤ηtr+r′xr′ U(x) , and the assertion then follows ast→ ∞sincer+r′<0.
• Proof of the sufficient condition of (ii)
Letr ∈R. Takingr′< −r, by hypothesis, there exists a constantx0>1 such that, for x≥x0, lim
t→∞tr′U(xt)
U(x) =0. Hence, forη>0, there exists a constantt0>1 such that, for t≥t0,tr′U(t x)≤ηU(x), or
U(t x)
(t x)r ≤η U(x) xrtr+r′.
Using the change of variabley=t xand noting thaty→ ∞ast→ ∞give, forr∈R, being r+r′>0,
y→∞lim U(y)
yr =0, which means thatU∈M∞.
• Proof of the necessary condition of (iii)
Letr∈Randη>0. Setr′> −r. SinceU∈M−∞there exists a constantx0>1 such that, forx≥x0,U(x)≥ηxr′. Hence, fort>1,
trU(t x) U(x) ≥η xr′
U(x)tr+r′, and the assertion then follows ast→ ∞sincer+r′>0.
• Proof of the sufficient condition of (iii)
Letr ∈R. Takingr′< −r, by hypothesis, there exists a constantx0>1 such that, for x≥x0, lim
t→∞tr′U(xt)
U(x) = ∞. Hence, forη>0, there exists a constantt0>1 such that, for t≥t0,tr′U(t x)≥ηU(x), or
U(t x)
(t x)r ≥ηU(x) xr t−r−r′.
Using the change of variabley=t xand noting thaty→ ∞ast→ ∞give, forr∈R, being
−r−r′>0,
y→∞lim U(y)
yr =0, which means thatU∈M∞.
Proof of Corollary 1.
LetU∈RV with tail indexρ. Then, fort>1,
x→∞lim trU(t x) U(x) =tr+ρ,
which implies that, forǫ>0, there exists a constantx0>1 such that, forx≥x0, tr+ρ−ǫ≤trU(t x)
U(x) ≤tr+ρ+ǫ.
Hence, settingτ= −ρ, gives, on the one hand, forr<τ,
−ǫ≤lim
t→∞trU(t x) U(x) ≤ǫ,
which implies lim
t→∞trU(t x)
U(x) =0 takingǫarbitrary, and, on the other hand, forr>τ,
t→∞limtrU(t x) U(x) = ∞.
Therefore one has, applying Theorem 1, thatU∈M withρU=ρ.
Finally, a function belonging toMbut not toRVis for instance the function given in Example 2.
Proof of Proposition 1.
• Proof of (i)
Letǫ>0. By definition ofU∈MwithρU= −τ, there exist constantsxa,xb>1 such that, forx≥xa,U(x)≤x−τ+ǫ, and, forx≥xb,U(x)≥x−τ−ǫ.
So, for x≥x0:=max(xa,xb),x−τ−ǫ≤U(x)≤x−τ+ǫ. Hence, for anyx0≤c<d < ∞, one has, settingMc:=min(c−τ−ǫ,d−τ+ǫ) andMd :=max(c−τ−ǫ,d−τ+ǫ), thatU satisfies Mc≤U(x)≤Mdfor anyx∈[c;d].
• Proof of (ii)
Letǫ>0. By definition ofU∈M∞, there exists a constantx0>1 such that, forx≥x0, U(x)≤xǫ. Hence, for anyc≥x0, one has, settingMc:=cǫ, thatUsatisfiesU(x)≤Mcfor anyx∈[c;∞).
• Proof of (iii)
Letǫ>0. By definition ofU∈M−∞, there exists a constantx0>1 such that, forx≥x0, U(x)≥xǫ. Hence, for anyd≥x0, one has, settingMd :=dǫ, thatUsatisfiesU(x)≥Md
for anyx∈[d;∞).
Proof of Theorem 2.
Letµbe the Lebesgue measure onR.
• Proof of (i)
LetU∈M withρU= −τand letr<τ. Applying Theorem 1, (i), there existsxa>1 such that, forx≥xa,
t→∞limtrU(t x) U(x) =0.
Letxa≤c<d< ∞. Then using Egoroff’s theorem (see e.g. [5]), there exists a measurable A⊆[c;d] of a positive measure such that
t→∞limsup
x∈AtrU(t x) U(x) =0.
Let us prove by contradiction that the previous limit holds on [c;d]. Then suppose that there existǫ>0,©
xnª
n∈N⊆[c;d], and© tnª
n∈N⊆R+such thattn→ ∞and
n→∞lim tnrU(tnxn)
U(xn) >ǫ. (11)
By Proposition 2 one has, denoting log(A)=©
log(x) :x∈Aª
and noting that log(A) has a positive measure,
µ³
n→∞lim(log(A)−log(xn))´
≥µ¡ logA¢
>0, which implies that there exist a constant log(u)∈Rand a subsequence©
xni
ª
i∈N⊆© xnª
n∈N such that log(xni)+log(u)∈log(A), i.e.u xni∈A. Note thatu>0.
By Proposition 1, (i), there exist 0<Mc≤Md< ∞such thatMc≤U(x)≤Md,x∈(c;d).
Hence, one then has
tnriU(tnixni) U(xni) =
µtni
u
¶rU³t
uniuxni
´
U(uxni) urU(uxni) U(xni) ≤
µtni
u
¶rU³t
niu uxni
´ U(uxni) urMd
Mc. Noting that
µtni
u
¶rU((tni
±u)uxni)
U(uxni) →0 sinceu xni∈Aandtni
±u→ ∞asni→ ∞provide tnriU(tnixni)
U(xni) →0 asni→ ∞, which contradicts (11).
• Proof of (ii)
LetU∈M withρU= −τand letr<τ. Applying Theorem 1, (i), there existsxb>1 such that, forx≥xb,
t→∞limtrU(t x) U(x) = ∞.
Letxb≤c<d< ∞and let© ǫmª
m∈Nbe a strictly increasing sequence of positive numbers such thatǫm→ ∞asm→ ∞. Then using Egoroff’s theorem, there exists a measurable Am⊆[c;d],m∈N, of a positive measure such that
t→∞lim inf
x∈Am
trU(t x) U(x) ≥ǫm. Let us prove
t→∞lim inf
x∈[c;d]trU(t x) U(x) = ∞ by contradiction. Then suppose that there existδ>0,©
xnª
n∈N⊆[c;d], and© tnª
n∈N⊆R+ such thattn→ ∞and
n→∞lim tnrU(tnxn)
U(xn) <δ. (12)
By Proposition 2 one has, denoting log(Am)=©
log(x) :x∈Amª
,m∈N, and noting that log(Am) has a positive measure,
µ³
nlim→∞(log(Am)−log(xn))´
≥µ¡ logAm¢
>0,
which implies, form∈N, that there exist a constant log(um)∈Rand a subsequence
©xnm,i
ª
i∈N⊆© xnª
n∈Nsuch that log(xnm,i)+log(um)∈log(Am), i.e. umxnm,i ∈Am. Note thatum>0 andc±
d≤um≤d±
c,m∈N.
By Proposition 1, (i), there exist 0<Mc≤Md< ∞such thatMc≤U(x)≤Mdforx∈(c;d).
Hence, one then has
tnrm,iU(tnm,ixnm,i) U(xnm,i) =
µtnm,i
um
¶rU³tnm,i
um umxnm,i
´
U(umxnm,i) umr U(umxnm,i) U(xnm,i) ≥ǫm
³c d
´r Mc
Md, implyingtnrm,iU(tnm,ixnm,i)
U(xnm,i) → ∞asm→ ∞, which contradicts (12).
• Proof of (iii)
LetU ∈M∞and letr∈R. Applying Theorem 1, (ii), there existsx0>1 such that, for x≥x0,
t→∞limtrU(t x) U(x) =0.
Letx0≤c<d< ∞.
On the one hand, by hypothesis, there exists a constantMd>0 such that, forx∈[1;d], U(x)≥Md. On the other hand, by Proposition 1, (ii), there exists a constantMc >0 such that, forx∈[c;∞),U(x)≤Mc. Combining these inequalities gives, forx∈[c;d], Md ≤U(x)≤Mc. Hence a proof similar to the one given to prove (i) can be done to conclude that lim
x→∞tr sup
x∈[c;d]
U(t x) U(x) =0.
• Proof of (iv)
LetU∈M−∞and letr∈R. Applying Theorem 1, (iii), there existsx0>1 such that, for x≥x0,
t→∞limtrU(t x) U(x) = ∞.
Letx0≤c<d< ∞.
On the one hand, by hypothesis, there exists a constantMd>0 such that, forx∈[1;d], U(x)≤Md. On the other hand, by Proposition 1, (iii), there exists a constantMc>0 such that, forx∈[c;∞),U(x)≥Mc. Combining these inequalities gives, forx∈[c;d], Mc≤U(x)≤Md. Hence a proof similar to the one given to prove (ii) can be done to conclude that lim
x→∞tr inf
x∈[c;d]
U(t x) U(x) = ∞.
Proof of Theorem CK.
LetU:R+→R+be a measurable function.
• Proof of (i)⇒(ii)
Letǫ>0 andU∈MwithρU=τ. One has, by definition, that
x→∞lim U(x)
xρ+ǫ=0 and lim
x→∞
U(x) xρ−ǫ= ∞.
Hence, there existsx0≥1 such that, forx≥x0,
U(x)≤ǫxτ+ǫ and U(x)≥1 ǫxτ−ǫ.
Applying the logarithm function to these inequalities and dividing them by log(x) (with x>1) provide
log (U(x))
log(x) ≤log (ǫ)
log(x)+τ+ǫ and log (U(x))
log(x) ≥ −log (ǫ) log(x)+τ−ǫ, and, one then has
x→∞lim
log (U(x))
log(x) ≤τ+ǫ and lim
x→∞
log (U(x)) log(x) ≥τ−ǫ,
from which one gets, takingǫarbitrary, τ≤ lim
x→∞
log (U(x)) log(x) ≤ lim
x→∞
log (U(x)) log(x) ≤τ, and the assertion follows.
• Proof of (ii)⇒(iii) Let 0<ǫ<1±
2. AssumeUsatisfies lim
x→∞
log (U(x))
log(x) =τ. Letγa measurable function with supportR+such thatγ(x)→0 asx→ ∞, and letb>1. Applying the L’Hôpital’s rule to the ratio gives
x→∞lim
γ(x)+ Rx
b log(U(s)) log(s) d s
s
log(x)
=lim
x→∞
log(U(x)) log(x) =τ First, supposeτ6=0, then
x→∞lim
log(U(x)) γ(x) log(x)+Rx
b log(U(s)) log(s) d s
s
=1,
and there existsx0>1 such that, forx≥x0, δU(x) := log(U(x))
γ(x) log(x)+Rx
b log(U(s)) log(s) d s
s
≥1−ǫ>0.
Settingx1:=max(b,x0) and defining the functions, forx≥x1,αU(x) :=γ(x)δU(x) log(x) andβU(x) :=log(U(x))±
log(x), the assertion follows.
Now, supposeτ=0. Define the functionV(x) :=xU(x),x>0, which clearly satisfies
x→∞lim
log(V(x))
log(x) =16=0. Hence, applying toV the previous proof forUwhenτ6=0 gives that there existx1,V ≥bV >1 and measurable functionsαV,βV, andδV satisfying, as x→ ∞,
αV(x)±
log(x)→0, βV(x)→1, δV(x)→1, such that, forx≥x1,V,
V(x)=exp
½
αV(x)+δV(x) Zx
bV
βV(s)d s s
¾ .
Defining, when τ=0, the constantx1,U :=x1,V and the functionsαU(x) :=αV(x)+ log(x)¡
δV(x)−1¢
,βU(x) :=βV(x)−1, andδU(x) :=δV(x), the assertion follows.
• Proof of (iii)⇒(i)
Suppose there existb>1 and measurable functionsα,β, andδsatisfying, asx→ ∞, α(x)±
log(x)→0, β(x)→τ, δ(x)→1, such that
U(x)=exp
½
α(x)+δ(x) Zx
b β(s)d s s
¾
, x≥x1for somex1≥b.
Letǫ>0 sufficiently small such that 2ǫ¡ τ+ǫ±
4¢
≤1 and 2ǫ¡ τ−ǫ±
4¢
≥ −1. Then there exist xa>1 such that, forx≥xa,¯
¯α(x)± log(x)¯
¯≤ǫ±
4,xb>1 such that, forx≥xb,¯
¯β(x)−τ¯
¯≤ ǫ±
4, andxc>1 such that, forx≥xc,¯
¯δ(x)−1¯
¯≤ǫ2/4.
On the one hand, writing, forx≥x0:=max(b,xa,xb,xc), U(x)
xτ+ǫ =exp
½
−(τ+ǫ) log(x)+α(x)+δ(x) Zx
b β(s)d s s
¾
=exp
½ log(x)
µ α(x) log(x)−ǫ
2
¶ +δ(x)
Zx
b β(s)d s s −
³ τ+ǫ
2
´log(x)
¾
≤exp
½
−ǫ
4log(x)+δ(x) Zx0
b β(s)d s s +δ(x)³
τ+ǫ 4
´¡
log(x)−log(x0)¢
−
³ τ+ǫ
2
´log(x)
¾
and noting that
δ(x)³ τ+ǫ
4
´
−
³ τ+ǫ
2
´
=¡
δ(x)−1¢³ τ+ǫ
4
´
−ǫ 4≤ −ǫ
8 give
x→∞lim U(x)
xτ+ǫ =0. (13)
On the one hand, writing, forx≥x0:=max(b,xa,xb,xc), U(x)
xτ−ǫ =exp
½
−(τ−ǫ) log(x)+α(x)+δ(x) Zx
b β(s)d s s
¾
=exp
½ log(x)
µ α(x) log(x)+ǫ
2
¶ +δ(x)
Zx
b β(s)d s s −³
τ−ǫ 2
´log(x)
¾
≥exp
½ǫ
4log(x)+δ(x) Zx0
b β(s)d s s +δ(x)³
τ−ǫ 4
´¡
log(x)−log(x0)¢
−³ τ−ǫ
2
´log(x)
¾
and noting that
δ(x)³ τ−ǫ
4
´
−
³ τ−ǫ
2
´
=¡
δ(x)−1¢³ τ−ǫ
4
´ +ǫ
4≥ǫ 8 give
xlim→∞
U(x)
xτ−ǫ = ∞. (14)
Combining (13) and (14) providesU∈M withρU=τ.
Proof of Proposition 4.
LetU:R+→R+be a measurable function.
AssumeU∈O-RVand the limit lim
x→∞
log(U(x))
log(x) exists. Applying Theorem CK givesU∈Mwith ρU= lim
x→∞
log(U(x)) log(x) .
Proof of Proposition 5. We will prove the proposition forλ= ∞. The proof forλ= −∞is sim- ilar.
Let us prove it by contradiction. Assume there existsU∈M∞TO-RV.
By assumptionU∈M, we have, forρ∈Randδ>0, there existsx0>1 such that, forx≥x0, U(x)≤cxρ. Applying the logarithm function to this inequality, dividing it by log(x),x>1, and taking the limitx→ ∞give
xlim→∞
log(U(x)) log(x) ≤ρ.
Takingρarbitrary provides
xlim→∞
log(U(x))
log(x) = −∞. (15)
Now, by assumptionU∈O-RV, applying Proposition 3, (i)⇒(ii), there existα,β∈Randx1>1, c>0 such that, for allt≥1 andx≥x1,
c−1tβ≤U(t x) U(x) ≤ctα.
Hence applying to these inequalities the logarithm function, dividing them by log(t),t>0, and taking the limitt→ ∞give
¯
¯
¯
¯lim
t→∞
log(U(t)) log(t)
¯
¯
¯
¯
≤max©
|α|,|β|ª
< ∞, which contradicts (15). The proposition is proved.
4 Conclusion
A new characterization of the classM introduced in [7], a strict larger class than the class of regularly varying functions (RV), was proved, and it was extended to the classesM∞and M−∞. This characterization together with other two given by Cadena and Kratz in [7] allowed the study of relationships betweenMand the well-known classO-RV, another extension of RV.
It was found that these classes satisfyM 6⊆O-RVandO-RV 6⊆M, and necessary conditions to have inclusions were provided. Relationships amongO-RVandM∞andM−∞were provided.
Note that any result obtained here can be applied to positive and measurable functions with finite support by using the change of variabley=1±
(xU∗−x) forx<xU∗wherexU∗is theendpoint ofUdefined byxU∗:=sup©
x:U(x)>0ª .
Acknowledgments
The author gratefully acknowledges the support of SWISS LIFE through its ESSEC research program on ’Consequences of the population ageing on the insurances loss’.
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